When you have a slow connection, the 0F-error can occure when there is a large incoming mail on the server. Use webmail to check this, with webmail you don't have to download the mail from the server. When there is a large mail in your mailbox, just save the attachment if you want to and delete the mail so you don't have to download it with your mailprogram anymore.

This is far from a new error, but in this case it's got nothing to do with the mail client. I'm getting reports left and right that since about 4am mail has not been able to be received. It can be sent, but not received.

The problem in this case is CPanel's pop server is not started up. What you have to do is restart cppop, either through WHM or type the following from the shell:

service cpanel stop
service cpanel start

Hopefully they'll fix whatever the problem was, looks like it was an upgrade.